Question: To calculate $31^2$, Emily mentally figures the value $30^2$ and adds 61. Emily subtracts a number from $30^2$ to calculate $29^2$. What number does she subtract?
We see that $29^2 = (30 - 1)^2 = 30^2 - 2\cdot 30 \cdot 1 +1 = 30^2 - 59$. Therefore, Emily subtracts $\boxed{59}$.